The area bounded by the curve 2x² + y² = 2 is

The given equation of the curve is:
$$2x^2+y^2=2$$

To identify the curve, we can rewrite the equation in standard form by dividing both sides of the equation by 2:
$$\frac{2x^2}{2}+\frac{y^2}{2}=\frac{2}{2}$$
This simplifies to:
$$\frac{x^2}{1}+\frac{y^2}{2}=1$$
Or in the standard form of an ellipse:
$$\frac{x^2}{1^2}+\frac{y^2}{{\left(\sqrt{2}\right)}^2}=1$$

This is the standard equation of an ellipse of the form:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
Comparing the two equations, we obtain the semi-axes of the ellipse:
$$a=1$$
and
$$b=\sqrt{2}$$

The standard formula for the area ($A$) enclosed by an ellipse is given by:
$$A=\pi ·a·b$$

Substituting the values of $a$ and $b$ into the area formula:
$$A=\pi ·1·\sqrt{2}$$
This simplifies to:
$$A=\sqrt{2}\pi \text{ sq. units}$$

Thus, the area bounded by the curve is indeed $\sqrt{2}\pi$ square units.